Generalized Absolute Stability Using Lyapunov Functions With Relaxed Positivity Conditions

Conditions are given for verifying stability and computing upper bounds on the induced (regional) L
₂
gain for systems defined by vector fields which are, along with their Jacobian, rational in the states and sector bounded nonlinearities. A class of candidate Lyapunov functions is considered that are polynomial on the states and the nonlinearities and have a polynomial scaled Lurie-Postnikov term. The main result of this letter is a set of conditions that relax the requirement on the candidate Lyapunov function from being sum-of-squares with respect to the nonlinearities and the Lurie-Postnikov terms from being non-negative.